A ball of radius 15 has a round hole of radius 5 drilled on an axis through its center. Find the volume of the resulting solid.

Question

poofypenguin · Answer

Here's my work so far:

poofypenguin · Answer

But the answer is coming out incorrect when I type it into a computer system.

poofypenguin · Answer

Can anyone please help me?

hartnn · Answer

method is correct, volume of sphere - vol of cylinder.
maybe type 3750pi units^3 
or try inserting value of pi

poofypenguin · Answer

I tried that but it didn't work :(

poofypenguin · Answer

Is it possible of looking at it as a rotation of a semicircle on a coordinate plane? I think that's what my teacher was hinting at, but I don't know how to do that... will that give a different answer?

hartnn · Answer

donno that either, but in any way you should get same volume...lets wait for other ppl's replies then...

unklerhaukus · Answer

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unklerhaukus · Answer

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unklerhaukus · Answer

polar coordinates?

agent0smith · Answer

Is it possible of looking at it as a rotation of a semicircle on a coordinate plane? This might be the way to go... Lets try using the washer method for volume: http://media.wiley.com/Lux/07/39807.nce019.gif $$x^2+y^2 = 15^2$$ so $$y = \sqrt[]{225-x^2}$$ Integrate from x=-sqrt200 to sqrt200 (found by entering y=5 since the drilled hole is radius 5). g(x) is 5 since we're subtracting the area or the rectangle under the semicircle. $$\pi \int\limits_{-\sqrt{200}}^{\sqrt{200}} \left[ (225-x^2)-5^2 \right]dx$$

agent0smith · Answer

Image will hopefully help. Area in blue is what we want, so subtract the rectangle in orange.

agent0smith · Answer

$$V = \pi \int\limits\limits_{-\sqrt{200}}^{\sqrt{200}} \left[ (225-x^2)-5^2 \right]dx = \pi \left[ 200x-x^3/3 \right]_{-\sqrt{200}}^{\sqrt{200}}$$

$$V = 3771.236 \pi $$units^3

Which appears pretty reasonable, considering your earlier answer. Your method gets a volume that's too small because the cylinder volume you subtracted is slightly too big.

agent0smith · Answer

You can also integrate from x = 0 to sqrt200 and then just double that answer, for easier calculations.

agent0smith · Answer

I wrote that at 5am and I assumed you knew a bunch of the stuff (eg how to write the equation of a circle, some knowledge on solids of revolution). If I lost you at some point, lemme know and I'll explain in more detail.

agent0smith · Answer

I hate writing out a nifty solution and nobody cares :P

agent0smith · Answer

@PoofyPenguin

poofypenguin · Answer

That's an awesome solution @agent0smith! That's exactly what I was looking for! I just could not figure it out! :D

Thanks so much @agent0smith !

agent0smith · Answer

No prob, you're welcome :)